Strong / Weak Edge Vertex Mixed Domination Number of a Graph
Radhakrishna Bhat, S. S. Kamath, Surekha R Bhat · 2012
For any v€ V, the set N(v)={u € V ! uv € E} is the openneighbourhood of the vertex v; while the set N[v]=N(v) U [v] is the closed neighbourhood of v. Similarly for any edge x=uv,N(x)= {y € E} y is adjacent to x } and N [x]=N(x)U{y}. further for any edge x=uv, Vx(N[x]) = {w € V! uw € E}. An edge x, m-dominates a vertex v if € Vx. Anedge x strongly (weakly) m –dominates the vertex v. A set L E is an Edge Vertex Dominating set (EVD-set) if every vertex in G is m-dominated by an edge in L. The edge vertex domination number Y (g) is the minimum cardinality of an EVD.set. A set L E is a strong edge vertex dominating set (SEVD-set) [Weak edge vertex dominating set (WEVD-set0] if every vertex in G is strongly (weakly) m-dominated by an edge in L. The strong (weak) edge vertex domination number Y(G) (y(G)) is the minimum cardinality of SEVD-set (WEVD-set), Besides finding the relationship between the existing graph parameters we prove a Gallai’s type results for edges. A new parameter called Edge Vertex Degree of an edge is defined and a bound in terms of the maximum and minimum EV degree is established.